One basic concept in mathematics that I see students struggle with, and with which I struggled myself, is the notion of expressions. However, when we remove the mathematical component, we can see that expressions behave much like a concept that natural language speakers deal with fairly easily.

So let’s start there, in natural language.

You can call me Paul, or you can call me Mr. Hartzer. Here are a bunch of things you can call me, in fact:

- Paul
- Mr. Hartzer
- Paul Hartzer
- Sir
- Hey, you
- Teacher
- Mr. H.
- That guy over there
- The man currently wearing a red t-shirt and navy slacks

My son calls me Daddy, but unless he’s around, it would be weird if you did (unless you’re him, in which case, “Hi, son! Glad you’re reading this post years after I wrote it!”).

The list above deserves two immediate comments. First, it’s not by any means exhaustive: Quite the contrary. There are probably an infinite number of ways you could describe me in a way that would identify me in a particular context. Secondly, some of those items are fairly dependent on me in general and not on my context (Paul, Mr. Hartzer, Paul Hartzer), while others are dependent on my context and could easily apply to others if the conditions change (that guy over there, sir).

Philosophically, we could note that the natural language referents are scalar: Some depend *more* on me than on context (not any random male is named Paul), and some depend *more* on context than on me, but none of them define me uniquely in all possible contexts. Anybody named “Paul Hartzer” can properly be identified by that name, and if there are two Pauls in the same context, referring to “Paul” is ambiguous.

However, doing so gets into the weeds a bit: The relevant points are

- Names (“Mr. Paul Hartzer”) describe a specific person or thing, while other natural language identifiers (“Hey, you”) describe whoever or whatever seems to fit the bill in a particular context;
- There are a wide variety of names and other identifiers that all describe the some specific person or thing in the same context.

The latter point forms the basis of understanding mathematical expressions. When considered at the level of Algebra I, expressions fall into two basic categories, analogs of the buckets described above:

- Dependent on the object itself: 4, 2+2, 2·2, 2
^{2}, 100_{2} - Dependent on the context: x+x, x·x, x
^{x}, 100_{x}

The first list always refer to the same thing. The second list refers to 4 when x refers to 2 (or in the case of x·x, when x refers to -2).

I think there’s a tendency to think in terms of all those other things equaling 4 and 4 itself being some inherent “thing”. The philosophical reality is that the “thing” is some abstract notion to which 4 happens to have been accepted as the simplest way to refer.

Oh, that was a mouthful. That’s what happens when we try to maintain the rigors of mathematical language.

Let’s go back to natural language for a moment. I’m not the name “Mr. Hartzer”. I’m a human being. As I describe above, when someone wants to refer to me, there are a bunch of ways they can do so. When I’m in school, if a student refers to me as “Mr. Hartzer” to anyone else familiar with the staff of the school, then everyone in the conversation knows who’s being discussed. Therefore, it’s simplest to refer to me as “Mr. Hartzer”.

If a student refers to me as “my fifth hour teacher”, that’s a less efficient way to refer to me even though it may be perfectly unambiguous. The student only has one fifth hour teacher, and presumably I’m that teacher, but anyone else in the conversation has to know *additional information* beyond the staff roster of the school to know who’s being referred to. That’s a less efficient way to refer to me.

“Mr. Hartzer” and “my fifth hour teacher” are *expressions* that refer to me, just as 4 and 2+2 and x+x (when x = 2) are *expressions* that refer to the concept of a set of objects: {object, object, object, object}. 4 happens to be accepted as the most efficient way to refer to that set (conventionally, because we use a base greater than 4: if binary was accepted as our default base, than we would call that set of objects 100 just as easily as we now call it 4), just as in school settings “Mr. Hartzer” happens to be accepted as the most efficient way to refer to me.

This isn’t a trivial point, even though it may seem like a difficult one to wrap your mind around. Technically speaking, when we say 2+2=4, we’re making two different statements: An arithmetic one, and one relating to expressions.

The arithmetic one is the one that elementary students are most familiar with. We’re saying that if we take a set of two objects and put it together with another set of two objects, we get a set of four objects. {object, object} + {object, object} == {object, object, object, object}. Of course, in elementary school, we start with sets of actual objects to make this clearer. *Johnny has two apples. Sally gives him two more apples. How many apples does he have now?*

The important transition to make in algebra is to realize that 2+2 and 4 are not themselves actual things, but rather that they’re ways of referring to the same thing (namely, a set of objects). They’re **equivalent expressions**. There are myriad other ways of referring to the same set of objects. The crucial point is that, just as I am a specific person but have many acceptable names, there is a specific notion of “a set of four objects” which likewise can be referred to in many, many ways.

Indeed, a crucial action in mathematics, the thing that makes our modern mathematical notation so powerful, is the way in which we can manipulate these equivalent expressions in order to detect patterns previous mathematicians couldn’t see.

For instance, let’s say we know that \(x + 2 = 4\). We don’t know \(x\), which is to say, we know that \(x\) represents the size of some set, but we don’t know how many objects are in that set. That’s what we want to find out.

If we realize that 4 and 2+2 are different names (expressions) for the same thing (the abstract set of four objects), we can swap out one name for the other to make the initial problem clearer: \(x + 2 = 2 + 2\). We can then remove the +2 from each side and get the value of \(x\).

To be sure, we can see the solution of \(x + 2 = 4\) exclusively in mathematical terms. But seeing the middle step in terms of replacing equivalent expressions (that is, names) instead of in mathematical terms may provide a different perspective that helps some students understand what’s going on.

A final point for this entry: My first list included “The man currently wearing a red t-shirt and navy slacks”. Likewise, we can come up with some fairly complex expressions that resolve to something simple. For instance, we could express the notion of 4 as: \[\int_1^{11_2} x dx\]

Not a simple expression, by any means, but mathematically equivalent (as an expression) to 4.